I'm interested in a very specific sub-set of this, which is barely mentioned in the answers to this other question.

What factors should I consider when choosing an aspect ratio? Obviously for photos I intend to print, I may wish to choose a standard size, or when fitting a photo into a specific spot in a magazine or web site, I need to choose a specific size/aspect ratio.

But given full artistic freedom, what aspect ratios are considered most pleasing, in what contexts, and why?

Honestly, I think what you are asking it too subjective. The best aspect ratio is the one that gives the best photo, but it could be anything.
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ItaiJul 14 '11 at 22:54

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@Itai: Many, many, many topics on this site are subjective. Such is the nature of art, no? Yet these other questions have good information in their answers, even if there is no single "right" answer.
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FlimzyJul 14 '11 at 23:19

None of those examples are as subjective as this one, IMO mostly because you have limited the question artistry only.
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ItaiJul 14 '11 at 23:58

The rule of thirds is all about artistry as well.
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FlimzyJul 15 '11 at 0:23

@Flimzy, I'm with you on artistry, but isn't this really covered under "what things should you take into account - both practical and artistic?" in the other question? (Emphasis mine.)
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mattdmJul 15 '11 at 2:42

Thanks for the examples with my photograph. Adding space on a certain side of the photograph can actually make you "think" about it. It adds some tension. I will put a small black border around them to better illustrate the effect.
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Nick BedfordAug 31 '11 at 22:34

Consider, that today the primary medium to view photos is screen, not paper. Unlike paper, computer or TV screen is almost always observed in landscape orientation, and it cannot be rotated (tablets and hand-held devices being an exception).

This means that landscape crops have an advantage, because they can use more screen space and can deliver more detail on the same screen. Also they seem to correspond better to human angle of view. Square and portrait crops should be used only when they are really important for the photo, if in doubt go landscape.

Popular screen aspect ratios are: 4:3 (e.g. 1024×768), 16:10 (e.g. 1280×800, 1440×900) and 16:9 (e.g. 1366×768, 1600×900, 1920×1080). There is a trend towards wider screens, so 16:10 and 16:9 formats (close to golden ratio, 1.618, by the way) are becoming more common. If FullHD becomes the gold standard, 16:9 may become the most used screen aspect ratio in few years. It is even wider than 3:2 format used in many DSLRs.

John Camp examined a "standard art text", which is a photobook of notable paintings, for size, for aspect ratio and for orientation (landscape/portrait). Though he used such a small sample size, his article convinced me and it answers your question. He reports,

Because of the history of the "golden rectangle," I went into this thinking that a lot of the paintings' aspect ratio would be about 1:1.61 or 1.62. I was wrong. Only one fell right on it, and four others, close. ... Forty-four out of the 72 had aspect ratios that fell between 1:1.2 and 1:1.5.

As for me, I stick to standard television's 1.33, unless the subject is really wide, such as a tableau vivant, or really tall, like a campanile. However, exceeding 1:1.5 guarantees it won't fit typical photo print sizes

Given complete freedom, the Golden Ratio has long been considered the ideal. Roughly, it is 1:1.618. This ratio has been use for thousands of years in photographs, magazines, architecture (going back to the Greeks) and other designs. Check the wikipedia page on this.

There is another ratio called the Silver Ratio which aproximages 1:1.414. This is used in things such as A4 paper.

A4 paper, like all the An papers, is 1:1.414 (i.e. sqrt(2)).
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Edgar BonetJul 15 '11 at 8:52

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Note that use in art/design for thousands of years is actually kind of dubious, even though it's widely believed to be the case. And, modern studies don't show much evidence for an innate aesthetic preference for this ratio either. See my answer here. That's not to say it's not a mathematically interesting ratio, though.
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mattdmJul 15 '11 at 13:06

This is the sort of answer I was looking for. Guidelines for a ratio that is aesthetically pleasing to the human eye. And @mattdm, I really like your answer you linked to--it has a lot of good, relevant information.
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FlimzyJul 15 '11 at 14:59

8x10 very popular size
9x12 model portfolios
5.5x8.5 comp cards
11x17
13x19
17x22 popular large size for home printers

All of these fit into the aspect range of 0.65-0.80. Note that many popular sizes, such as 8x10 and 17x22 are significantly more "square" than the standard 35mm frame. This is to be kept in mind when composing.

For years I was puzzled by 13x19.... until I had to matt prints for a gallery. A 3:2 gets a nice 0.5" border all around while printed at 12x18". The matt ends up covering most of the border but the framing remains intact.
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ItaiJul 15 '11 at 1:24

I personally only crop when, aesthetically, it has a lot to do with the subject or the limitations of having to simply take the photograph with the 3:2 aspect ratio with the camera. For example, I took a photo of a mate of mine in landscape when I could have, in hindsight, tilted the tripod mount to portrait and taken it, but because I was in the shoot myself I didn't really think of that at the time. Luckily, with an 18mp sensor, cropping is not going to do much to the end result.

So given there was literally nothing but a white wall on either side of the subject, I cropped to a square and I was happy with the result. In the end, the photo was to be used as an online profile picture anyway so the square crop worked in its favour.

Landscapes are another genre where cropping may help to enhance the impact of the photograph. If you have taken a expansive photo of some mountains, for example, and didn't think of taking a series of panoramic shots, it might help to enhance the mood of the photo by doing a more cinematic widescreen crop, such as anamorphic 2.35:1 or even just 16:9. This can remove uninteresting parts of the scene such as a gradient sky and help to put more focus on the vast mountain ranges.

The Bottom Line

Many photographers perceive cropping an outright disgrace, some have no problem with it and some are on the fence, only cropping when there will be a tangible benefit from doing so.

I tend to fall in the latter category. A 3:2 photo is one that can be printed easily and I also believe a 3:2 photo is one that fits in with our human vision's aspect ratio (180 degrees horizontal, 120 degrees vertical). It feels right, but that's not to say that cropping is in no way useful. It most certainly is needed for many situations.